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The central assumption of calculus is that you can divide infinitely. If you can't, the operations fall apart.

They just become approximations rather than precisely accurate.

well, doesn't the math become rather horrible though? Like, rather than f(x)=sin(x) f''= -sin(x)ddx (whatever the correct notation for that is) you get some massive but finite series of terms that just approximate it?

If you don't assume that it works, then you can't move on and do other math without resorting to very complicated geometry. It's pretty much why calc was invented.

Calc isn't just great because you can use it to model things acutely. It's great because you can precisely model things using very simple computations.

I'd like to see more technical computer education. People should be able to install parts and operating systems, set up a network, have some idea about what the what different parts of the computer and operating system do, and have an awareness of security and privacy issues. Some legal stuff too would probably be a good idea, like copyright and licensing. Regardless of where kids end up, these kind of things will be valuable skills, and it's sure something I was never taught formally.

redx on October 2008

This machine kills threads.

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FeralThat's what I do. I drink, and I know things.Registered Userregular

The central assumption of calculus is that you can divide infinitely. If you can't, the operations fall apart.

They just become approximations rather than precisely accurate.

well, doesn't the math become rather horrible though? Like, rather than f(x)=sin(x) f''= -sin(x)ddx (whatever the correct notation for that is) you get some massive but finite series of terms that just approximate it?

If you don't assume that it works, then you can't move on and do other math without resorting to very complicated geometry. It's pretty much why calc was invented.

Calc isn't just great because you can use it to model things acutely. It's great because you can precisely model things using very simple computations.

Well, basically, yeah. Otherwise you're using Riemann sums which are fine for a computer but godawful for a human. The point I was trying to make is that when you use calculus on discrete data rather than continuous data, you end up with a very very close (in some cases infinitesimally close) approximation of the truth. I mean, if you're calculating the surface area of a sphere, the error caused by the fact that your curve is made up of atoms instead of infinitely small points is completely irrelevant for most applications.

Calculus may not reflect reality 100%, but that does not make it invalid. It is a rigorously constructed system that can easily stand alone or be applied. Even though matter is discrete, we can still often approximate it, on useful scales, by using calculus. QM is built upon a thorough base of calculus and linear algebra. There isn't much use in trying to push away calculus' power in giving us a good idea of reality.

And see, I can go the rest of my life not knowing any of this and continue to be a bitchin' translator.

Yeah.

Like I said before, I think algebra, geometry, intro to trig, and a half-year of statistics should be our basic educational standards.

Algebra and geometry because they're useful in daily life. Want to retile your bathroom or recarpet your living room? Being able to calculate the area of irregular shapes would definitely help. Want to figure out how much money you'd need to bring in to support a child? Algebra helps with that.

Intro to statistics should be necessary so you can at least have a good idea of the difference between 'significant' and 'statistically significant' and so you can have slightly better than a snowball's chance in hell of identifying bogus statistics when they're used to prop up bad politics or snake oil products sold in infomercials.

I'm still not convinced "whether or not I'm going to use this in the field" is a reliable metric of what should or should be not learned at any given stage of education, especially in high school. High school is about establishing, at least in theory, the bare minimum every citizen should be familiar with, because the smarter we are as a society, the better off we are. College ought, obviously, to tend more towards specialization, but there still needs to be some way of figuring out what exactly all "educated" people should know.

Now, Calculus in particular? I'm not going to defend that specific subject, but we cannot dismiss it merely because we are unlikely to apply it. That's already been established as a poor way of measuring whether or not something should be taught. I do accept the argument that the disproportionate emphasis on math tends to take away from other general education requirements. But a better basis for whether or not something should be taught needs to be established.

The central assumption of calculus is that you can divide infinitely. If you can't, the operations fall apart.

String theory is not the most credible theory. The most credible theory is the standard model, if you're defining "credible" as having the most experimental support.

String Theory is patently untestable, and, for the most part, unverifiable with current super-colliders.

And you just don't understand calculus at all. I can't really think of a nicer way to put that D:

Edit: To make that less out-right insulting: Scalfin, what other mathematical tool would you use to describe the world other than calculus?

If your idea of a "real" mathematical device is something that perfectly describes reality, then you fundamentally misunderstand math and science. In that sense, all of modern human thought, including science and math, is an abstraction, and just as useless to you as you think calculus is.

And see, I can go the rest of my life not knowing any of this and continue to be a bitchin' translator.

Yeah.

Like I said before, I think algebra, geometry, intro to trig, and a half-year of statistics should be our basic educational standards.

Algebra and geometry because they're useful in daily life. Want to retile your bathroom or recarpet your living room? Being able to calculate the area of irregular shapes would definitely help. Want to figure out how much money you'd need to bring in to support a child? Algebra helps with that.

Intro to statistics should be necessary so you can at least have a good idea of the difference between 'significant' and 'statistically significant' and so you can have slightly better than a snowball's chance in hell of identifying bogus statistics when they're used to prop up bad politics or snake oil products sold in infomercials.

I pretty much agree with that. I think knowing a bit of the concepts of Calculus is a good way of expanding your braining, but it's by no means particularly important to grasp in everyday life. Basic mathematics definitely is, though. Stats in particular should be drilled into everyone's head at least a bit. The amount of BS marketers and politicians can get away with if they have a pretty graph is unconscionable.

I don't know that many schools require Calc, though. I think my local high school had the option of AP Calc, Calc, or College Algebra. Might have been another class for really math-uninterested people.

I suppose I'm biased, but I really think everyone should at least be exposed to differential equations, which, obviously, requires calculus.

Fuckin everything is differential equations. Maybe I'm alone at watching things happening around me and trying to explain those behaviors mathematically, but differential equations, and calculus, for that matter, are an incredible analytical tool. You can be a great linguist, or even the best construction worker ever, but I dare you to claim a solid fundamental understanding of the world around you without ever seeing the language of limits.

Calculus isn't hard goddamnit! It's ruined by bad professors, weed-out courses, and a crappy secondary education. It's ruined by the material being vomited on students who either don't want it or are even incapable of seeing it for what it is by nature of the way in which it's presented.

Of course everyone doesn't require calculus in day-to-day life. I'll even concede many people will never have a direct use for it. But knowing it--understanding it,--I'm convinced, is an intellectual hurdle for truly appreciating the beauty of mathematics. And it is beautiful--but so much moreso than it is a hurdle: show me a "difficult" calculus class and I'll show you a professor that didn't care whether his students understood it or not--I'll even show you students that didn't care to try something just for their own sake.

I'd like to see more technical computer education. People should be able to install parts and operating systems, set up a network, have some idea about what the what different parts of the computer and operating system do, and have an awareness of security and privacy issues. Some legal stuff too would probably be a good idea, like copyright and licensing. Regardless of where kids end up, these kind of things will be valuable skills, and it's sure something I was never taught formally.

I don't think it need necessarily go as far as being able to build computers or even set up networks, but yeah, understanding of computers is something that should definitely be taught. Is it not already in US schools? When I went to school (over 11 years ago now), Computing was a mandatory class for the first four years of high school. We were taught all about input, process, output; taught to use office software such as spreadsheet software (even using formulas to create a theatre seating plan in a spread sheet), CAD software etc. It was also an optional class for Highers, where they taught programming (Pascal when I took it). I can only imagine that the curriculum is now more advanced, especially covering internet, email and likely privacy, security and copyright laws. There was also a separate secretarial studies class that everyone had too take where they taught everyone touch typing and that sort of thing. It wasn't a permanent class, just a module that they cycled with other modules such as a Library class where they taught you how to use the Dewy Decimal system etc.

I dunno, maybe the US educational system is a bit antiquated. I get the impression from this discussion that the British system has a more balanced approach between arts, languages (typically most schools will make a foreign language mandatory or highly advised up until Highers or A Levels), maths, sciences and vocational studies. But then people are always on the news complaining about how it's been 'dumbed down' because they don't teach Latin etc. any more. I suspect maybe it's just been made 'more relevant' (Knowing how to use and understanding computers is arguable considerably more relevant and valuable than being able to read a redundant language).

I suppose I'm biased, but I really think everyone should at least be exposed to differential equations, which, obviously, requires calculus.

Fuckin everything is differential equations. Maybe I'm alone at watching things happening around me and trying to explain those behaviors mathematically, but differential equations, and calculus, for that matter, are an incredible analytical tool. You can be a great linguist, or even the best construction worker ever, but I dare you to claim a solid fundamental understanding of the world around you without ever seeing the language of limits.

Calculus isn't hard goddamnit! It's ruined by bad professors, weed-out courses, and a crappy secondary education. It's ruined by the material being vomited on students who either don't want it or are even incapable of seeing it for what it is by nature of the way in which it's presented.

Of course everyone doesn't require calculus in day-to-day life. I'll even concede many people will never have a direct use for it. But knowing it--understanding it,--I'm convinced, is an intellectual hurdle for truly appreciating the beauty of mathematics. And it is beautiful--but so much moreso than it is a hurdle: show me a "difficult" calculus class and I'll show you a professor that didn't care whether his students understood it or not--I'll even show you students that didn't care to try something just for their own sake.

You are biased.

I think everyone would have frequent application for, and benefit greatly, from learning more linguistics than we currently do. This involves doing things like studying Latin like Quid said, maybe taking a course or two on language development among children and among second language learners. I think knowing those things would help the average American greatly when living in our multicultural society.

Could I learn calculus if I tried? Yes.
Could I apply calculus once I learned it? Yes.
Do I have actual necessity for calculus in my every day life? No.
Could I spend my time more effectively doing something besides learning calculus? Yes.

We're by no means denigrating the value of knowing calculus. We're simply saying that it shouldn't be a requirement in everyone's education (which it, or another replacement math course, was at my universities).

Of course everyone doesn't require calculus in day-to-day life. I'll even concede many people will never have a direct use for it. But knowing it--understanding it,--I'm convinced, is an intellectual hurdle for truly appreciating the beauty of mathematics. And it is beautiful--but so much moreso than it is a hurdle: show me a "difficult" calculus class and I'll show you a professor that didn't care whether his students understood it or not--I'll even show you students that didn't care to try something just for their own sake.

I took Calculus twice (and passed both times--I was trying to get a better grade the second time.) "The beauty of mathematics" is not a phrase I would ever apply towards calculus. It was hard, it was boring, and I hated it. And now, years after I took calculus, I don't remember any of it because I never used it in my job (or for anything else.) By contrast, I remember a lot from the advanced literature classes, which I loved. I got so much out of those classes . . . but that doesn't mean everyone should take them or that they, too, would get a lot out of them.

I'd like to see more technical computer education. People should be able to install parts and operating systems, set up a network, have some idea about what the what different parts of the computer and operating system do, and have an awareness of security and privacy issues. Some legal stuff too would probably be a good idea, like copyright and licensing. Regardless of where kids end up, these kind of things will be valuable skills, and it's sure something I was never taught formally.

I don't think it need necessarily go as far as being able to build computers or even set up networks, but yeah, understanding of computers is something that should definitely be taught. Is it not already in US schools? When I went to school (over 11 years ago now), Computing was a mandatory class for the first four years of high school. We were taught all about input, process, output; taught to use office software such as spreadsheet software (even using formulas to create a theatre seating plan in a spread sheet), CAD software etc. It was also an optional class for Highers, where they taught programming (Pascal when I took it). I can only imagine that the curriculum is now more advanced, especially covering internet, email and likely privacy, security and copyright laws. There was also a separate secretarial studies class that everyone had too take where they taught everyone touch typing and that sort of thing. It wasn't a permanent class, just a module that they cycled with other modules such as a Library class where they taught you how to use the Dewy Decimal system etc.

I dunno, maybe the US educational system is a bit antiquated. I get the impression from this discussion that the British system has a more balanced approach between arts, languages (typically most schools will make a foreign language mandatory or highly advised up until Highers or A Levels), maths, sciences and vocational studies. But then people are always on the news complaining about how it's been 'dumbed down' because they don't teach Latin etc. any more. I suspect maybe it's just been made 'more relevant' (Knowing how to use and understanding computers is arguable considerably more relevant and valuable than being able to read a redundant language).

It's district by district with typically high principal autonomy equaled out by schedules made rigid by complexity (My school had an unusually robust student legislature, and we were told not to bother because we'd never be able to accommodate enough people in the new schedule for it to work).

Could I learn calculus if I tried? Yes.
Could I apply calculus once I learned it? Yes.
Do I have actual necessity for calculus in my every day life? No.
Could I spend my time more effectively doing something besides learning calculus? Yes.

We're by no means denigrating the value of knowing calculus. We're simply saying that it shouldn't be a requirement in everyone's education (which it, or another replacement math course, was at my universities).

Note that I further qualified it with differential equations in particular--it's not a necessity, it's just a way to look at things--a different, more accurate, more complete, highly developed way to analyze information. I suppose if you don't really want to know how the world works, you can neglect calculus entirely, but isn't that one of the foundations of education?

You could go all semantics right there and claim that I'm arguing before Leibniz and Newton, no one knew how the world worked--that no one could live their lives. Practically, yes, you can live a productive life never knowing calculus--I'm just arguing that you're shooting yourself in the foot, intellectually, if you don't.

I took Calculus twice (and passed both times--I was trying to get a better grade the second time.) "The beauty of mathematics" is not a phrase I would ever apply towards calculus. It was hard, it was boring, and I hated it. And now, years after I took calculus, I don't remember any of it because I never used it in my job (or for anything else.) By contrast, I remember a lot from the advanced literature classes, which I loved. I got so much out of those classes . . . but that doesn't mean everyone should take them or that they, too, would get a lot out of them.

I can't really argue with you here, because I absolute hate how calculus is taught at universities. In fact, I take issue with the teaching methods for pretty much all math at all levels short of select undergraduate (after calculus 1 and 2) and graduate courses.

Math teachers aren't english teachers: this becomes a huge problem when clear communication is the difference between understanding something instantly, and hating it forever because you find it useless.

On your second point, I'd have to disagree. I think everyone would get something out of advanced literature if they actually put time into it. This all falls back on relative "difficulty" of courses, and, in my opinion, the fear built in to the U.S. education system to fail people.

At my High School, there were people that graduated with no hours of college credit, and there were people that graduated with 50+ hours of college credit. Beyond the immediate question of why this disparity exists, why do we even allow it? Don't you agree that this implies the baseline "Gen Ed" is ridiculously low for High Schools? Why not require advanced literature in High School? Because some people will fail if we "raise" standards? Because a lot of people will fail?

So what if a lot of people fail? Shuffling criminally unprepared people into college, telling them they must be an engineer to make money-to be successful, and watching them change majors to underwater basketweaving by the droves is not a societally successful education model.

It just isn't plain economical. Establish the "general education" in High School--thoroughly, and then let people spend tens of thousands of dollars on an education for which they're prepared.

Teachers stigmatize reading by forcing it on kids. It shouldn't be "Read it or you fail, you worthless ward of the state!", it should be "Read it because it tastes like cannndddyyyy!"

Of course, that's akin to asking teachers to make kids love to read that already don't, which takes a very gifted teacher.

One of the dirty secrets I learned hanging out with former high school English teachers in grad school is that the majority of teachers hate the books they teach. In a lot of districts, reading lists are standardized and books are chosen for their pedantic value (The Scarlet Letter = Symbolism!). This leads to unenthusiastic teachers forcing books they despise on unenthusiastic students.

Of course, the foundation of our education system in our test-happy era is unhappy teachers forced to teach rigid lesson plans to uninterested students.

Could I learn calculus if I tried? Yes.
Could I apply calculus once I learned it? Yes.
Do I have actual necessity for calculus in my every day life? No.
Could I spend my time more effectively doing something besides learning calculus? Yes.

We're by no means denigrating the value of knowing calculus. We're simply saying that it shouldn't be a requirement in everyone's education (which it, or another replacement math course, was at my universities).

Note that I further qualified it with differential equations in particular--it's not a necessity, it's just a way to look at things--a different, more accurate, more complete, highly developed way to analyze information. I suppose if you don't really want to know how the world works, you can neglect calculus entirely, but isn't that one of the foundations of education?

You could go all semantics right there and claim that I'm arguing before Leibniz and Newton, no one knew how the world worked--that no one could live their lives. Practically, yes, you can live a productive life never knowing calculus--I'm just arguing that you're shooting yourself in the foot, intellectually, if you don't.

Again -- I get that knowing more is always a good thing. I'm an English major who primarily uses algebra and computer science to do my daily job (but don't be fooled -- my communication skills do wonders for me being able to further apply those skills as well).

What I'm saying is that calculus does not have enough universal application in specialized fields of study to warrant requiring every student to take it. Students on math or science tracks should definitely study it, but students specializing in other fields really shouldn't be required. They can take it if they want, and they'll probably benefit from it, but they don't need to.

And again -- the original point is that there is too great an emphasis placed on college track academics, despite not only there being a good number of students who have no interest in those field of studies, we need students who don't have an interest in those fields of study, who choose to specialize in other things. Not everyone can be a computer programmer. Not everyone can be a marine biologist. Not everyone can be a translator. Not everyone can be a hedge fund account manager. etc. so on and so forth.

We need people who aren't on the college track of education, but as our education system is currently set up, those people are short-changed and under-served.

Again -- I get that knowing more is always a good thing. I'm an English major who primarily uses algebra and computer science to do my daily job (but don't be fooled -- my communication skills do wonders for me being able to further apply those skills as well).

What I'm saying is that calculus does not have enough universal application in specialized fields of study to warrant requiring every student to take it. Students on math or science tracks should definitely study it, but students specializing in other fields really shouldn't be required. They can take it if they want, and they'll probably benefit from it, but they don't need to.

And again -- the original point is that there is too great an emphasis placed on college track academics, despite not only there being a good number of students who have no interest in those field of studies, we need students who don't have an interest in those fields of study, who choose to specialize in other things. Not everyone can be a computer programmer. Not everyone can be a marine biologist. Not everyone can be a translator. Not everyone can be a hedge fund account manager. etc. so on and so forth.

We need people who aren't on the college track of education, but as our education system is currently set up, those people are short-changed and under-served.

Now that I can absolutely agree with. My desire for people to learn calculus is really more tertiary to my desire for people to learn more things/choose a career path sooner.

College is glorified, and it's painted as the one road to the American Dream. This is a patent lie, and it bothers me that our system is structured that way.

And as much as I may want everyone to know calculus, I absolutely realize and concede that that's completely infeasible, but I won't concede that people will be better off if they did.

High school is often very early for people to start to specialize. Heck, there are Freshmen in college who have no idea what their majors will be.
I think the focus on trade schools after high school, with everybody getting a well-rounded high school education might be a better idea, especially to help prevent people from being tracked into the "trade path" very early due to either apathy or background/socioeconomics. Prepare everybody to learn as much as they can in order to help them be as much as they can be (sorry to be so hokey there.)
I don't use 90% of what I learned in high school in my everyday life. This includes most of my humanities classes. But I am glad I took them, and glad I had to take a ridiculous number of humanities classes in college, because it made me "well rounded" even though I was a science major.
Higher math is needed so much in college level math/science, that you really do start at a disadvantage if you are not at least taking Calc I your first semester. I don't see that calculus is needed for every high school student, heck, I didn't take it in high school and I became a math major in college. But the mathematical thinking required for higher level math does help in logical thinking and figuring out things in this world. I became frustrated having to prove basic mathematics in my advanced calc class in college, because it was stuff every middle schooler knew how to do. But knowing how to break down the problem and prove it from basic principles really helped me develop. So while I knew the math easily enough, the process of proving the math and the steps one takes does help your thinking.
Some calculus teachers might say "memorize this crap", and they are the worst teachers ever.My pre-calc and calc teachers taught the basic principles behind the math, and why and how it was developed. This helps you to understand how it is useful in the world. This critical thinking is, to me, the point behind math. And then you can use it as a tool once you've understood it.
I also like having students learn a foreign language, because oftentimes English becomes less about the basics of grammar and you really learn grammar better (imo) when you are looking at language from a whole other perspective. Latin was my choice, but I honestly think reading a shit-ton of books helped my vocab as much f not more. The grammar really stuck with me, though.
I also have other opinions on other subjects, but I am writing too much.
So, I think high school students should be challenged to take as many classes as they can in subjects they may not always use. It not only allows them to change their mind and do something else with their life if they chose to later, but it also helps them develop. Then later they can move onto further education or career of their choice when they've got a few things under their belt.